What are the properties that make circles similar?

Question

At first, when thinking about similar circles I tried extending the properties of similar triangles to circles to reason that all circles are similar.

The properties of similar triangles are as follows:

1) Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal).

2) Corresponding angles must have the same measure.

Each side of a triangle must be scaled up or down by the same factor to get a "new" triangle that has proportional corresponding sides to the "original" triangle.

I figured a similar thing must happen between similar circles. If the radius of a circle is scaled up or down by some factor then its circumference will also be scaled up or down by that same factor. The "new" circle formed will be a scaled up or down version of the "original" circle, and its radius and circumference will be proportional to the "original's" radius and circumference. In fact, the ratios between their respective circumferences and radii would both simplify to $2\pi$.

Any two circles, regardless of size, will always have the same central angle of $360^\circ$.

Therefore, I concluded that all circles are similar by extending the properties of similar triangles.

Granted, this isn't a rigorous way of defining nor proving the similarity between all circles, but is my "intuition"/rough reasoning correct?

Also, what are the properties of similar circles? Are they at all akin to the properties of similar triangles?

Answer 1

If you want to define similarity of general shapes, not just polygons, you have to be quite rigorous.

The standard way is to simply list all possible transformations which intuitively don't affect similarity, and then define two things to be similar if they are connected by such a transformation.

In standard Euclidean geometry, these transformations are usually taken to be translations, uniform scalings, mirrorings, rotations, and any (finite) combination of such.

So any two circles are similar because you can translate and scale one to make it into the other.

Answer 2

There is no sense in using properties of a triangle to prove (or convince yourself) that all circles are similar. Start with definition of similarity:

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection $f$ from the space onto itself that multiplies all distances by the same positive real number $r$, so that for any two points $x$ and $y$ we have

$$d ( f ( x ) , f ( y ) ) = r d ( x , y ),$$

where "$d(x,y)$" is the Euclidean distance from $x$ to $y$. The scalar $r$ has many names in the literature including: the ratio of similarity, the stretching factor and the similarity coefficient. When $r = 1$ a similarity is called an isometry (rigid motion).

Given two circles are you able to find a bijection which transforms one circle into another?

Hint: a translation followed by a dilation is the way to go.

If you proceed this way you find out that the property that makes all circles similar is the fact, that a circle is the set of all points (in a plane) that are at a given distance (the radius) from a given point (the center).

Answer 3

As a high school mathematics teacher (who also happens to have studied graduate-level mathematics),

Blockquote I figured a similar thing must happen between similar circles. If the radius of a circle is scaled up or down by some factor then its circumference will also be scaled up or down by that same factor. The "new" circle formed will be a scaled up or down version of the "original" circle, and its radius and circumference will be proportional to the "original's" radius and circumference. ... Any two circles, regardless of size, will always have the same central angle of 360∘.

... is a fine explanation of why all (two-dimensional) circles are similar.

Common Core Standard

In fact, I supposedly required to teach a similar 'proof' in my high school geometry classes.

So to summarize, in the coordinate plane, dilations create similar figures (same shape, different size) and translations create congruent figures. Since any two circles in the coordinate plane can be mapped to one another by finding the correct dilation and a sequence of translations, all circles (or at least all circles in the coordinate plane) are similar. Tranformations are one way to 'define' the concept of congruence - it's honestly the approach I like more with a typical teenager. To extend the concept to a non-coordinate plane, I would assume (but haven't Google'd) that it has been demonstrated that there exists a sequence of straight-edge & compass constructions to accomplish a similar feat.